b-super-imposed codes were introduced in 1964 by W. H. Kautz and R. C. Singleton [9], and the concept of bd-super-imposed codes were introduced by A. J. Macula [12]. As stated in Section 2.2 a bd-disjunct matrix is a bd-super-imposed code in matrix language. The bd-disjunct matrix can be used to construct an error-tolerable design for non-adaptive group testing, which has applications to the screening of DNA sequence, and the corresponding decoding algorithm is efficient. See [3], [6] for details.
A bd-disjunct matrix is also called a pooling design.
The constructions of bd-disjunct matrices were given by many authors, e.g. [11], [12], [13], [4]. Theorem 2.2.3 is a special case of [7]. The algorithm in Theorem 2.3.2 was given in [6]. See [4] for more results of this line of study.
3
Pooling spaces
We constructed disjunct matrices from the lattice of subsets of a given set in Theorem 2.2.3. We generalize the idea to poset in this chapter.
3.1 Preliminaries
We now give the basic definitions and properties of a partially ordered set. The expert may want to skip the remaining of this section and go to the next section.
Let P denote a finite set. By a partial order on P, we mean a binary relation ≤ on P such that
(i) x ≤ x ∀ x ∈ P,
(ii) x ≤ y and y ≤ z −→ x ≤ z ∀ x, y, z ∈ P, (iii) x ≤ y and y ≤ x −→ x = y ∀ x, y ∈ P.
By a partially ordered set (or poset, for short), we mean a pair (P, ≤), where P is a finite set, and where ≤ is a partial order on P. By abusing notation, we will suppress reference to ≤, and just write P instead of (P, ≤).
Let P denote a poset, with partial order ≤, and let x and y denote any elements in P. As usual, we write x < y whenever x ≤ y and x 6= y, and write x 6< y whenever x < y is not true. We say y covers x whenever x < y, and there is no z ∈ P such that x < z < y. A poset can be described by a diagram in which the elements are corresponding to dots, and y covers x whenever dot y is placed above dot x with an edge connecting them. See Fig. 1 for the diagram of the poset with five elements {0, w, x, y, z}, and w, x covers 0; y covers w, x; z covers w, x respectively. Note 0, w, y is a direct chain of length 2.
c
c c
c c
@@
@
¡¡¡
©©©©©© HH HH HH
0
w x
y z
Figure 1. A poset.
An element x ∈ P is said to be minimal (resp. maximal) whenever there is no y ∈ P such that y < x (resp. x < y). Let min(P ) (resp. max(P )) denote the set of all minimal (resp. maximal) elements in P. Whenever min(P ) (resp. max(P )) consists of a single element, we denote it by 0 (resp. 1), and we say P has the least element 0 (resp. the greatest element 1).
Throughout the chapter 2 we assume P is a poset with the least element 0. By an atom in P, we mean an element in P that covers 0. We let AP denote the set of atoms in P. By a rank function on P, we mean a function rank from P to the set of nonnegative integers such that rank(0) = 0, and such that for all x, y ∈ P, y covers x implies rank(y) − rank(x) = 1. Observe the rank function is unique if it exists. P is said to be ranked whenever P has a rank function. In this case, we set
rank(P ) := max{rank(x)|x ∈ P },
Pi := {x|x ∈ P, rank(x) = i},
and observe P0 = {0}, P1 = AP. Observed P is ranked if and only if for any x ∈ P every direct chain from 0 to x has the same length.
Let P denote any finite poset, and let S denote any subset of P. Then there is a unique partial order on S such that for all x, y ∈ S, x ≤ y in S if and only if x ≤ y in P. This partial order is said to be induced from P. By a subposet of P, we mean a subset of P, together with the partial order induced from P. Pick any x, y ∈ P such that x ≤ y. By the interval [x, y], we mean the subposet
[x, y] := {z|z ∈ P, x ≤ z ≤ y}
of P.
P is said to be atomic whenever for each element x of P, x is the join of atoms in the interval [0, x]. Suppose P is atomic and x < y are two elements in P . Observe the atoms in the interval [0, x] is a proper subset of atoms in the interval [0, y].
Let P denote any poset, and S be a subset of P. Fix z ∈ P. Then z is said to be an upper bound (resp. lower bound) of S, if z ≥ x (resp. z ≤ x) for all x ∈ S.
Suppose the subposet of upper bounds (resp. lower bounds) of S has a unique minimal (resp. maximal) element. In this case we call this element the least upper bound or join (resp. the greatest lower bound or meet) of S. If S = {x1, x2, . . . , xt} we write x1∨ x2∨ · · · ∨ xt for the join of S and x1∧ x2∧ · · · ∧ xt for the meet of S. P is said to be meet semi-lattice (resp. join semi-lattice) whenever P is nonempty, and x ∧ y (resp. x ∨ y) exists for all x, y ∈ P. A meet semi-lattice (resp. join semi-lattice) has a 0 (resp. 1). A meet and join semi-lattice is called a lattice.
Suppose P is a lattice. Then P is said to be upper semi-modular (resp. lower semi-modular ) whenever for all x, y ∈ P,
y covers x ∧ y −→ x ∨ y covers x (resp. x ∨ y covers x −→ y covers x ∧ y).
P is said to be modular whenever P is upper semi-modular and lower semi-modular.
3.2 Definitions
Now we can give the main definition of the chapter as following.
Definition 3.2.1. Let P be a ranked poset. For any w ∈ P, define w+ = {y ≥ w | y ∈ P }.
P is said to be a pooling space whenever w+ is atomic for all w ∈ P.
In particular, a pooling space is atomic. It is immediate from the definition that if P is a pooling space, then so is w+ for any w ∈ P. The following theorem is a generalization of Theorem 2.2.3.
Theorem 3.2.2. Let P be a pooling space with rank D ≥ 1. Fix an element x ∈ PD and fix an integer b (1 ≤ b ≤ D). Let T ⊆ PD be a subset such that |T | ≤ b and x 6∈ T. Then there exists an element y ∈ [0, x] ∩ Pb such that y z for all z ∈ T.
Proof. We prove the theorem by induction on D. If D = 1 then b = 1 and the theorem holds by setting y = x. In general, pick an element z ∈ T. Then x 6= z by assumption.
Since x is the least upper bound of [0, x] ∩ P1 and x 6≤ z, z is not an upper bound of [0, x] ∩ P1. Hence we can pick an element w ∈ [0, x] ∩ P1 such that w 6≤ z. Then T ∩ w+ has at most b − 1 elements. In the pooling space w+, the element x and the elements of T ∩ w+ all have rank D − 1, and the elements of w+∩ Pb have rank b − 1.
Hence by induction, we can choose y ∈ [w, x] ∩ Pb such that y 6≤ u for all u ∈ T ∩ w+. Note that clearly y 6≤ u for all u ∈ T \ w+. This proves the theorem.
3.3 The contractions of a graph
Many examples of pooling spaces were given in [7]. These are related the Hamming matroid, the attenuated space, and six classical polar spaces. Among these examples there is a common property: each interval is modular. In this section we will construct pooling spaces without modular intervals. Throughout the section let G denote a simple connected graph on n vertices.
Definition 3.3.1. Let P = P (G) denote the set of partitions A of the vertex set V (G) such that the subgraph induced by each block of A is connected. For A, B ∈ P , define
A ≤ B ⇐⇒ A is a refinment of B.
The poset (P (G), ≤) is called the poset of contractions of G.
Example 3.3.2. Let G denote a graph with the vertex set {w, x, y, z} and edge set {wx, xy, yz, zw}, i.e. G is the 4-cycle C4. Then the poset P (G) is as in Fig. 2. We delete the single element blocks in the notation of a partition. e.g. the notation 0 is used to denote the partition with four blocks {w}, {x}, {y}, {z}, and wx is used to denote the partition with three blocks {w, x}, {y}, {z}. The poset is a lattice, but not a modular lattice. This is because the join of the elements wx yz and xy zw is wxyz, which covers wx yz, but xy zw does not covers the element 0 which is the meet of the elements wx yz and xy zw. Observe the subposet induced on wx+ is P (C3), the poset of contractions of a triangle.
c that are contained in the same block of B. Let C be an element in P (G) that glues these two blocks of A. Then A < C ≤ B and rank(C) = rank(A) + 1. This shows C = B and rank(B) = i + 1.
Theorem 3.3.4. P (G) is a pooling space of rank n − 1.
Proof. P (G) is ranked by previous lemma. From previous lemma and the definition each atom in P(G) contains n − 1 blocks, one block containing two adjacent vertices and each of the remaining n − 2 blocks containing a single vertex. By identifying the atoms with the edges of G we find each element A ∈ P (G) is the join of those edges contained in the subgraph of G induced by A. This shows that P (G) is atomic. More generally, for B ∈ P (G), the poset B+ is also atomic. This is because the subposet B+ is isomorphic to the poset P (BG) of contractions of BG, where BG is the graph with the vertex set B, and for two distinct blocks x, y ∈ B x is adjacent to y whenever some vertex in x is adjacent to some vertex in y.
Remark 3.3.5. Let G = Kn denote the complete graph on n vertices. Then the elements in P = P (Kn) are all the partitions of the vertex set of Kn. S(n, k) := |Pk| is called the Stirling number of the second kind. It is well known that S(n, k) can be solved by the recurrence relation
S(n, k) = S(n − 1, k − 1) + kS(n − 1, k) for 1 ≤ k ≤ n − 1
with initial condition S(n, 0) = 0 for n ≥ 1, and S(n, n) = 1 for n ≥ 0. See [2, Section 8.2] for details.
3.4 Finite fields
Before going farther, we need some background on finite fields. Recall that a finite field Fq is a set of q elements containing 0,1 with two binary relations + , · , such that (Fq , + , 0) and (Fq∗ , · , 1) are abelian groups, and +, · satisfy distribute law, where Fq∗:=Fq− {0}.
We give some examples as following.
Example 3.4.1. {0, 1, 2, 3} is not a finite field under ususal + , · (mod 4), since 2 does not have the multiplication inverse.
Example 3.4.2. F4 = {0, 1, x, x + 1} is a finite field under + , · (mod x2+ x + 1).
It is well-known that the finite field Fq of q elements is unique up to isomorphism, and q = pr for some prime p. There are two ways to describe Fq:
(i) Fq = {a0+ a1x + a2x2+ · · · + ar−1xr−1 | ai ∈ Zp}, (ii) Fq = {0, 1, γ, γ2, · · · , γq−2}.
The + defined in (i) is as usual, and · is defined mod some irreducible polynomial g(x) ∈ Fq[x] of degree r, e.g. g(x) = x2 + x + 1 in Example 3.4.2. The · defined in (ii) is as usual with the condition γq−1 = 1 and the + is defined mod g(x). γ is called a primitive element of Fq.
Example 3.4.3. F4 = {0, 1, x, x + 1} = {0, 1, x, x2} (mod x2+ x + 1).
Example 3.4.4. F5 = {0, 1, 2, 3, 4} = {0, 1, 2, 22, 23} (mod 5).
Note 3.4.5. Fq is the set of solutions of x(xq−1− 1) = 0.
Note 3.4.6. Suppose q = pr for some prime p. Then Fq is a vector space over Fp. Lemma 3.4.7. Suppose T ⊆ Fpm is a subspace over Fp. Then γT is a subspace over Fp for any γ ∈ Fpm.
Proof. This is clear for γ = 0. Suppose γ 6= 0, and suppose α1, α2, . . . , αk is a basis of T. Then γα1, γα2, . . . , γαk is a basis of γT.
3.5 Projective and affine geometries
We introduce two more examples of pooling spaces in this section.
Definition 3.5.1. The projective geometry P G(n, q) is the poset consisting of all subspaces of Fqn with order defined by inclusion. The elements in Pi are referred to the i-subspaces of Fqn for i = 0, 1, 2, · · · , n.
The following is from linear algebra.
Note 3.5.2. dim(U + V )+dim(U ∩ V )=dim(U)+dim(V ) for U, V ∈ P G(n, q).
Definition 3.5.3. Consider the n-dimensional space Fqnwhere q is a prime or a prime power. Let
n k
q
denote the number of k-subspaces of Fqn. In convention, define
n k
q
= 0, if k > n or k < 0.
We list a few properties for
n k
.
Lemma 3.5.4.
Proof. We prove the statement by induction on k.
n 0
q
= 1 is clear since {0} is the only one subspace of dimension 0,
and
since there are qn− 1 nonzero vectors in Fqn and each 1-subspace containing q − 1 nonzero vectors.
In general, by counting the number of pairs (W, V ), where W ⊆ V are (k − 1)-subspaces, k-subspaces respectively in two ways, we find
Proof. By Lemma 3.5.4,
The following theorem will be used in the next section to construct super-imposed codes.
Theorem 3.5.7. Fix integers 0 ≤ r < k ≤ n. Let A, A1, A2, . . . , Ab be distinct k-subspaces of Fqn. Then there are at least
d := qk−r
r-subspaces of A which are not contained in each Ai for i = 1, 2, · · · , b.
Proof. To obtain the maximum elements of r-subspaces in A∩Ai, we assume dim(A∩
Ai)=k − 1 for all i = 1, 2, · · · , b. If A ∩ Ai 6= A ∩ Aj, then (A ∩ Ai) + (A ∩ Aj) = A
Hence b ≤ qr+ 1.
Suppose r = 1. Then
d > 0 ⇐⇒ b − 1 < q
⇐⇒ b ≤ q.
Note 3.5.9. Since
b ≤ q, r=1;
b ≤
2 1
q
= q + 1, r≥2 , we can choose A, A1, A2, · · · , Ab such that A ∩ Ai 6= A ∩ Aj for i 6= j, dim(A∩Ai)=k − 1 for every i=1,2,· · · , b and their meet is a (k − 2)−subspace. Then there are exactly d r-subspaces of A which are not contained in any Ai for i = 1, 2, · · · , b and d is defined in (3.5.1).
Now we consider the relation of projective geometry.
Definition 3.5.10. Let Fqn denote an n-dimensional vector space over a finite field Fq, where q is the number of elements in the field. Let P = P (Fqn) denote the poset with element set
P = {u + W | u ∈ Fqn and W ⊆ Fqn is a subspce} ∪ {∅},
where ∅ denote the empty set. The order is defined by inclusion. Note that P is a geometric lattice of rank n + 1. P is called the affine geometry and is denoted by AG(n, q). The elements in Pi are referred to the affine (i − 1)-subspaces of Fqn for i = 1, 2, · · · , n + 1. We say the affine subspaces u + W and v + W are parallel for u, v ∈ Fqn, W ⊆ Fqn is a subspace.
We immediately have the following lemma.
Lemma 3.5.11. Suppose u1, u2 ∈ Fqn and W1, W2 ⊆ Fqn are subspaces. Then u1 + W1 = u2+ W2 if and only if W1 = W2 and u1− u2 ∈ W1.
Now we have a similar version of Theorem 3.5.7
Lemma 3.5.12. Let A denote an affine k-subspaces of Fqn. Then the number of affine r-subspaces contained in A is
qk−r
where r < k. These affine r-subspaces in A are partitioned into
classes, each class consisting of qk−r parallel affine subspaces.
Theorem 3.5.13. Fix integers 1 ≤ r < k ≤ n. Let A, A1, A2, . . . , Ab be distinct affine k-subspaces of Fqn. Then there are at least
d := qk−r
affine r-subspaces contained in A, some of them in some affine subspace A ∩ Ai for each i = 1, 2, · · · , b to be deducted. A ∩ Ai takes maximal coverage of these affine r-subspaces when A∩Ai is an affine (k−1)-subspace, and in this situation the number of these affine r-subspaces is
q(k−1)−r
Corollary 3.5.14. In Theorem 3.5.13, if 0 < r < k2, then b = qr+1 is the largest integer such that d > 0; if r = 0, then b = q − 1 is the largest integer such that d > 0.
Proof. d > 0 ⇐⇒
b < q
k r
q
/
k − 1 r
q
= q · (qk− 1)(qk−1− 1) · · · (qk−r+1− 1) (qk−1− 1)(qk−2− 1) · · · (qk−r− 1)
= q(qk− 1) (qk−r− 1)
= qk+1− q − qk+1+ qr+1
qk−r− 1 + qr+1
= q(qr− 1)
qk−r− 1 + qr+1 Since
0 < r < k 2, Then
q(qr− 1) qk−r− 1 < 1.
Hence 0 < b ≤ qr+1. Suppose r = 0. Then
d > 0 ⇐⇒ b < q
⇐⇒ b ≤ q − 1
Note 3.5.15. Since
b ≤ q − 1, r=0;
b ≤ q, r≥1 and k ≤ n, we can choose Ai to be an affine k−subspace with the meet with A corresponding to each of the q parallel affine (k − 1)−subspaces in A. Then there is exactly d affine r−subspaces contained in A and not contained in any of Ai for i = 1, 2, · · · , b and d is defined in (3.5.2).
3.6 Codes on projective and affine geometries
We are clearly to apply the results in the section 3.5 to construction of super-imposed codes as following.
Definition 3.6.1. Let Pq(n, k, r) denote the incidence matrix of the set of r-subspaces and the set of k-subspaces in Fqn for 1 ≤ r ≤ k ≤ n. The following corollary is immediate from Theorem 3.5.7, Corollary 3.5.8 and Note 3.5.9.
Corollary 3.6.2. The columns of Pq(n, k, r) form a bd-super-imposed code, but not a bd+1-super-imposed code, where b is a positive integer satisfying
b ≤ q, r=1;
b ≤ q + 1, r≥2, k ≤ n and d is defined in (3.5.1).
Definition 3.6.3. Let Aq(n + 1, k + 1, r + 1) denote the incidence matrix for of the set of affine r-subspaces and the set of affine k-subspaces in Fqn 0 ≤ r ≤ k ≤ n. The following Corollary is immediate from Theorem 3.5.13, Corollary 3.5.14 and Note 3.5.15.
Corollary 3.6.4. The columns of Aq(n + 1, k + 1, r + 1) form a bd-super-imposed code, but not a bd+1-super-imposed code, when b is a positive integer satisfying
b ≤ q − 1, r=0;
b ≤ q, r≥1, k ≤ n and d is defined in (3.5.2).
We set r = 0 and b = q − 1 to obtain the following result.
Corollary 3.6.5. Let Aq(3, 2, 1) be the incidence matrix of the set of affine 0-subspaces and the set of affine 1-subspaces in Fq2. Then the columns of Aq(3, 2, 1) are (q − 1)1
-super-imposed code. ¤
3.7 Sperner’s theorem and EKR theorem
We list two interesting classical theorems in this section as following.
Theorem 3.7.1. (Sperner’s Theorem)Let M be an n × s 1-disjunct matrix. Then
s ≤ are n! maximal chains in P . Observe there are k!(n − k)! maximal chains containing a fixed x ∈ P with |x| = k. Observe for any chain L. |L ∩ F | ≤ 1. By counting the
Theorem 3.7.2. (EKR-Theorem) Let A be a collection of s distinct k-subsets of {1, 2, · · · , n}, where k ≤ n2, with the property that any two of the subsets have a
nonempty intersection. Then
s ≤
n − 1 k − 1
.
Proof. For a permutation σ of {1, 2, · · · , n}, and T ∈ A, define σ(T ) := {σ(x)|x ∈ T } and Aσ := {σ(T )|T ∈ A}. Set Si := {i, i + 1, · · · , i + k − 1} mod n for i = 1, 2, · · · , n and F := {S1, S2, · · · , Sn}. Observe for each Si ∈ F, there are 2k − 1 Sj ∈ F with Si∩ Sj 6= ∅. These are Si−(k−1),Si−(k−2),· · · ,Si, Si+1,· · · ,Si+k−1. Divide these into k boxes {Si−(k−1), Si+1},{Si−k−2, Si+2},· · · , {Si−1, Si+k−1},{Si}. Any two in the same boxes have empty intersection. Hence we can choose only one. From this observation we have |A ∩ F | ≤ k. Also |Aσ ∩ F | ≤ k for any permutation σ. We count (S, T, σ) in two ways, where S ∈ F , T ∈ A, σ is a permutation with σ(T ) = S, S ∈ Aσ ∩ F and T = σ−1(S), in the orders S,T ,σ and σ,S,T to find
n · s · k!(n − k)! ≤ n! · k.
Hence
s ≤ (n − 1)!
(k − 1)!(n − k)! =
n − 1 k − 1
.
Definition 3.7.3. Let P be a ranked poset of rank n and 1 ≤ k ≤ n be an integer.
We say P has the kth EKR property whenever any family F ⊆ Pk such that for any x, y ∈ F there exists a 6= 0 with a ≤ x and a ≤ y, we always have |F | ≤ |w+∩ Pk| for some w ∈ P1.
Conjecture 3.7.4. EKR property holds on a geometric lattice.
3.8 Remarks
The name pooling spaces was given in [7]. Theorem 3.3.4 was proved in [8]. Theo-rem 3.5.7 was given in [4] with a minor correction. TheoTheo-rem 3.5.13 was given in [8].
Theorem 3.7.1 and Theorem 3.7.2 are well known and have many different proofs.
We follow the proofs from [10, Chapter 6].
4
Reed-Muller Codes
For the remaining of the thesis, we consider the codes defined with more algebraic aspect, but it turns out these codes also have combinatorial meaning.
4.1 Linear Codes
Definition 4.1.1. A code C ⊆ Fqn is a [n, k, d]-linear code (or [n, k]-linear code) if C is a subspace of Fqn with dimension k and minimum distance d.
Definition 4.1.2. For any x ∈ C, the weight wt(x) of x is the number of nonzero coordinates in x. The minimum weight wt(C) of C is
wt(C) := min{w(x) | x ∈ C, x 6= 0}.
In general the weight of an element in Fqn depends on how the basis is chosen.
In the above definition the weight is associated with the standard basis of Fqn. We might choose different basis and define the weight differently. Because the distance of codewords have relation with the weight.
Note 4.1.3. The distance ∂(x, y) between the codeword x and y is wt(x − y) for any x, y ∈ C.
Note 4.1.4. We say C is a linear code if and only if x − y ∈ C and αx ∈ C for any x, y ∈ C and scalar α.
Note 4.1.5. If C is linear code, then the weight wt(C) is equal to the minimum distance d(C).
Note 4.1.6. The concept of weight of a code indeed depends on the chosen basis of vector space.
4.2 Reed-Muller Codes
At first, we give the definition of the codes considered in this chapter.
Definition 4.2.1. We define Rm :={f | f : F2m −→ F2 is a function}, where Rm is called the Reed-Muller code of order m.
The following two notes are clear.
Note 4.2.2. The Reed-Muller code is a vector space under usual +,· operations of functions.
Note 4.2.3. The Reed-Muller code of order m is a vector space over F2 of dimension 2m and |Rm| = 22m.
We consider a few special functions in Rm.
Definition 4.2.4. For 1 ≤ i ≤ m, we define xi ∈ Rm such that for any u ∈ F2m, xi(u) = 1⇐⇒ui = 1, and define 1 ∈ Rm such that for any u ∈ F2m, 1(u) = 1.
Definition 4.2.5. xi1xi2· · · xij∈ Rmis called a monomial of degree j where 1 ≤ j ≤ m and 1 ≤ i1, i2, · · · , ij ≤ m are distinct integers. 1 is called a monomial of degree 0.
We identify 0,1,2,· · · ,2m− 1 with the elements in F2m by using binary expressions, e.q. 0 = (0, 0, · · · , 0), 1 = (1, 0, · · · , 0, 0), 2 = (0, 1, 0, · · · , 0, ),· · · . We choose a
standard basis f0,f1,· · · ,f2m−1 of Rm, where fi(j) = 1 if and only if j = i for 0 ≤ i ≤ 2m− 1. We use the standard basis to express the codeword f ∈ Rm, so the weight of f has the following meaning.
Note 4.2.6. Suppose the function f ∈ Rm. Then f2 = f and the weight wt(f ) is equal to |f−1(1)|.
We consider the weight of a monomial as following note.
Note 4.2.7. Suppose f = x1x2· · · xr.Then
f−1(1) = {(1, 1, · · · , 1, ar+1, ar+2, · · · , am) | ai = 0 or 1}
is a affine (m − r)-subspace of F2m. Hence wt(x1x2· · · xr) = 2m−r. We find a basis of Rm.
Theorem 4.2.8. The set of monomials with degree less or equal m forms a basis of the Reed-Muller code of order m.
Proof. There are
m 0
+
m 1
+ · · · +
m m
= 2m monomials and dim(Rm) = 2m. It suffice to show monomials span Rm. Suppose f ∈ Rm. Observe
f = X
a∈f−1(1)
Ym j=1
(xj + aj + 1).
Hence f is spanned by monomials.
We consider Reed-Muller codes in the light of monomials.
Definition 4.2.9. RM (r, m):={f ∈ Rm | f is spanned by monomials of degree ≤ r}
where r ≤ m. RM (r, m) is called the r-th Reed- Muller Code of order m. Let wtm denote the weight function on RM (r, m).
From Theorem 4.2.8 and Definition 4.2.9, we have
Note 4.2.10. Since RM(r, m) is a linear code with codewords of length 2m, the dimension is dimRM(r, m) =
m 0
+
m 1
+ · · · +
m r
.
Theorem 4.2.11. The minimum distance d(RM (r, m)) is equal to 2m−r. Proof. We have seen
wtm(x1x2· · · xr) = 2m−r. Hence d(RM (r, m)) ≤ 2m−r. We prove
d(RM (r, m)) ≥ 2m−r by induction on m. Suppose m = 1.
Case 1: m = 1, r = 0. f : F21 −→ F2(no xi appears) and f = 1. Hence f−1(1) = F2. Then wt1(f ) = |f−1(1)| = 2 = 2m−r.
Case 2: m = 1, r = 1. f 6= 0 has wt1(f ) ≥ 1 = 2m−r.
Suppose for any 0 6= f ∈ RM (r, m), we have wtm(f ) ≥ 2m−r. Choose any f ∈ RM (r, m + 1). Say f = g + xm+1h where g ∈ RM (r, m + 1) without xm+1 and h ∈ RM (r − 1, m + 1) without xm+1.
Case 1: g = h 6= 0. Then f = h(xm+1) and
wtm+1(f ) = wtm(h) ≥ 2m−(r−1) = 2m+1−r. (Using h has at most r − 1 variables).
Case 2: g 6= h. Then
wtm+1(f ) = wtm(g) + wtm(g + h).
(To assign xm+1 = 0 in wtm(g) and xm+1 = 1 in wtm(g + h)).
Case 2.1: g = 0. Hence h 6= 0 and
wtm+1(f ) = wtm(h) ≥ 2m−(r−1) = 2m+1−r.
Case 2.2: g 6= 0. Note g + h 6= 0, since g 6= h. Hence
wtm+1(f ) = wtm(g) + wtm(g + h) ≥ 2m−r+ 2m−r = 2m+1−r.
Next, our goal is to prove
wtm(fS) = 2m−r ⇐⇒ S is affine (m−r)−subspace (∗) where S ⊆ F2m, and
fS(x) :=
1, if x ∈ S;
0, else fS is called the characteristic function of S.
Remark 4.2.12. Rm={fS | S ⊆ F2m}.
One direction is easier.
Theorem 4.2.13. Suppose S is an affine (m − r)-subspace in F2m. Then wt(fS) = 2m−r and fS ∈ RM (r, m).
Proof. Note wt(fS)=|fS−1(1)|=|S|=2m−r. Observe S is the solution space of a system of r linear independent equations in m variables. Hence there exist aij, bi ∈ F2 such that for i = 1, 2, · · · , r and j = 1, 2, · · · , m we have
(x1, x2, · · · , xm) ∈ S ⇐⇒
Xm j=1
aijxj = bi for i = 1, 2, · · · , r.
Observe
fS = Yr i=1
[(
Xm j=1
aijxj) − bi+ 1]
and the degree of the monomial in the expansion of fS is less or equal r.
To prove the other direction, we need some facts as following notes.
Note 4.2.14. An affine k-subspace is the union of 2 parallel affine (k − 1)-subspaces by Lemma 3.5.12.
Note 4.2.15. We say the disjunct union S1˙∪S2 = S ⊆ F2mif and only if fS = fS1+fS2. Theorem 4.2.16. The vectors in {fS | S is a affine (m − r)-subspace of F2m} span RM(r, m).
Proof. It suffices to prove xi1xi2· · · xit is spanned by the characteristic function of affine (m − r)-subspaces, where t ≤ r. Observe xi1xi2· · · xit=fT for some affine (m − t)-subspace T and fT = fT1+ fT2 for some parallel affine (m − (t + 1))-subspaces T1,T2. Keeping doing this, we find xi1xi2· · · xit is the sum of some characteristic functions of affine (m − r)-subspaces.
Definition 4.2.17. An affine (m − 1)-subspace in F2m is called a hyperplane of F2m. Theorem 4.2.18. Suppose T ⊆ F2m with |T | = 2k. Suppose |T ∩ S| = 0, 2k−1 or 2k for any hyperplane S of F2m. Then T is an affine k-subspace of F2m.
Proof. We prove this by induction on m and m = 2 is clear. In general, we consider the following 3 cases.
Case 1: T ⊆ S for some hyperplane S of F2m. Then S ∼= F2m−1. Let H be a hyperplane of S. Then H is an affine (m − 2)-subspace of F2m. We want to show that |T ∩ H| = 0, 2k−1 or 2k. Observe there is an affine (m − 1)-subspace S0 such that S ∩ S0 = H. Hence |T ∩ H| = |T ∩ S ∩ S0| = |T ∩ S0| = 0, 2k−1 or 2k by assumption. By induction, T is an affine k-subspace in S and then in F2m. Case 2: T ∩S = ∅ for some hyperplane S of F2m. Then T ⊆ S0 for the hyperplane S0 of F2m parallel to S. So the result follows from Case 1.
Case 3: |T ∩ S| = 2k−1 for all hyperplanes S of F2m. Observe the case m = k is clear, so suppose m 6= k. Then on the one hand
X and on the other hand
X where the summations are over all hyperplanes S in F2m. Hence
m = k,
a contradiction.
Now we can show the other direction in (∗).
Theorem 4.2.19. Let f ∈ RM (r, m) be the minimum weight vector. Then f = fS
and then apply Theorem 4.2.18 to say S is an affine (m − r)-subspace. Observe F2m = H ∪ H0 where H0 is parallel to H. Observe fH, fH0 ∈ RM (1, m) by Theorem
4.2.16 and 1 = fH + fH0, since H ∩ H0 = ∅. Hence f fH, f fH0 ∈ RM (r + 1, m). By Theorem 4.2.11,
wt(f fH) = 0 or ≥ 2m−(r+1) and
wt(f fH0) = 0 or ≥ 2m−(r+1). Since
2m−r = wt(f )
= wt(f fH + f fH0)
= wt(f fH) + wt(f fH0), We have wt(f fH) = 0, 2m−r−1 or 2m−r. Hence
|S ∩ H| = 0, 2m−r−1 or 2m−r.
4.3 Decoding
We study the decoding of Reed-Muller codes in this section, we need the following
We study the decoding of Reed-Muller codes in this section, we need the following